Exotic instantons and duality Marco Bill Dip. di Fisica T eorica, - - PowerPoint PPT Presentation

Exotic instantons and duality

Marco Billò

• Dip. di Fisica T

eorica, Università di T

• rino

and I.N.F .N., sez. di T

• rino

15-th European Workshop on String Theory ETH, Zurich - September 8, 2009

Foreword

Mostly based on

• M. Billo, L. Ferro, M. Frau, L. Gallot, A. Lerda and
• I. Pesando, “Exotic instanton counting and heterotic/type

I’ duality,” JHEP 0907 (2009) 092, arXiv:0905.4586 [hep-th].

• M. Billo, M. Frau, L. Gallot, A. Lerda and I. Pesando,

“Classical solutions for exotic instantons?,”, JHEP 03 (2009) 056, arXiv:0901.1666 [hep-th]. It builds over a vast literature

◮ I apologize for missing references...

Plan of the talk

1

Introduction and motivations

2

“Exotic” instantons in type I’

3

Interpretation as 8d instanton solutions

4

The effective action

5

Conclusions and perspectives

Introduction and motivations

Non-perturbative sectors

in brane-worlds

◮ (Susy) gauge and matter sectors on the uncompactified

part of (partially wrapped) D-branes

◮ gauge couplings involve 1/gs × different volumes → string

scale not tied to 4d Planck scale

◮ chiral matter, families from multiple intersections,...

D7b CY3 D7a R1,3

Non-perturbative sectors

in brane-worlds

◮ (Susy) gauge and matter sectors on the uncompactified

part of (partially wrapped) D-branes

◮ gauge couplings involve 1/gs × different volumes → string

scale not tied to 4d Planck scale

◮ chiral matter, families from multiple intersections,...

D7b CY3 R1,3 E3a D7a

◮ Non-perturbative sectors from partially wrapped

E(uclidean)-branes

◮ Pointlike in the R1,3 space-time: “instanton configurations” ◮ Tractable in String Theory, with techniques in rapid

development

Ordinary instantons

W.r.t. the gauge theory on a given D-brane stack,

CY3 R1,3 E3a D7a

◮ E-branes identical to D-branes in the internal directions:

gauge instantons

◮ ADHM from strings attached to the instantonic branes Witten, 1995; Douglas, 1995-1996; ... ◮ non-trivial instanton profile of the gauge field Billo et al, 2001 ◮ Rules and techniques to embed the instanton calculus in

string theory have been constructed

Polchinksi, 1994; Green-Gutperle, 2000, ...; T urin/Rome/Münich/UPenn/Madrid,...

Exotic instantons

W.r.t. the gauge theory on a given D-brane stack,

CY3 R1,3 D7a E3c

◮ E-branes different from D-branes in internal directions do

not represent gauge instantons; they are called exotic or stringy instantons

◮ May explain important terms in the effective action:

neutrino Majorana masses, moduli stabilizing terms, . . .

Blumenhagen et al 0609191; Ibanez and Uranga, 0609213; ... ; ◮ Exponentially suppressed but not just exp(−1/g2), can

involve volumes of different internal cycles

◮ Need to understand their status in the gauge theory and

to construct precise rules for the “exotic” instanton calculus

Our strategy

◮ Select a simple example: D(-1)/D7 in type I’ theory,

sharing many features of stringy instantons

◮ Investigate the field-theory interpretation of D(-1)’s in

this 8d gauge theory

Billo et al, 2009a;

◮ Compute the non-perturbative effective action on the

D7’s extending the rules of stringy instanton calculus to this “exotic” case.

◮ Check against the results in the dual Heterotic SO(8)4

• theory. Impressive quantitative check of this string

duality.

Billo et al, 2009b

◮ Apply the technology to tractable example leading to 4d

models

Work in progress, T urin + T

• r Vergata

“Exotic” instantons in type I’

A D(-1)/D7 system in type I’

◮ T

ype I’ is type IIB on a two-torus T2 modded out by Ω = ω (−1)FL I2 where ω = w.s. parity, FL = left-moving fermion #, I2 = inversion on T2

◮ Ω has four fixed-points on T2 where

four O7-planes are placed

O7

A D(-1)/D7 system in type I’

◮ T

ype I’ is type IIB on a two-torus T2 modded out by Ω = ω (−1)FL I2 where ω = w.s. parity, FL = left-moving fermion #, I2 = inversion on T2

◮ Ω has four fixed-points on T2 where

four O7-planes are placed

◮ Admits D(-1), D3 and D7’s transverse

to T2

O7

A D(-1)/D7 system in type I’

◮ T

ype I’ is type IIB on a two-torus T2 modded out by Ω = ω (−1)FL I2 where ω = w.s. parity, FL = left-moving fermion #, I2 = inversion on T2

◮ Ω has four fixed-points on T2 where

four O7-planes are placed

◮ Admits D(-1), D3 and D7’s transverse

to T2

◮ Local RR tadpole cancellation requires

8 D7-branes at each fix point

D7

A D(-1)/D7 system in type I’

◮ T

ype I’ is type IIB on a two-torus T2 modded out by Ω = ω (−1)FL I2 where ω = w.s. parity, FL = left-moving fermion #, I2 = inversion on T2

◮ Ω has four fixed-points on T2 where

four O7-planes are placed

◮ Admits D(-1), D3 and D7’s transverse

to T2

◮ Local RR tadpole cancellation requires

8 D7-branes at each fix point

We focus on one fix point

The gauge theory on the D7’s

◮ From the D7/D7 strings we get N = 1 vector multiplet in

d = 8 in the adjoint of SO(8):

• Aμ, Λα, ϕm
• ,

μ = 1, . . . 8 , m = 8, 9

◮ Can be assembled into a “chiral” superfield

Φ(x, θ) = ϕ(x) +

• 2 θΛ(x) +

1 2 θγμνθ Fμν(x) + . . . where ϕ = (ϕ9 + iϕ10)/

• 2.

◮ Formally very similar to N = 2 in d = 4

Effective action on the D7

(tree level)

◮ Effective action in Fμν and its derivatives: NABI

Back

S = S(2) + S(4) + S(5) + · · · = 1 8πgs

• d8x

Tr

• F2

(2π)4α′2 − t8Tr

• F4

3(2π)2 + α′L(5)

• F, DF
• + · · ·

Effective action on the D7

(tree level)

◮ Effective action in Fμν and its derivatives: NABI

Back

S = S(2)+S(4) + S(5) + · · · = 1 8πgs

• d8x

Tr

• F2

(2π)4α′2 − t8Tr

• F4

3(2π)2 + α′L(5)

• F, DF
• + · · ·
• ◮ The quadratic Yang-Mills term S(2) has a dimensionful

coupling g2

YM ≡ 4πgs(2π

• α′)4

Effective action on the D7

(tree level)

◮ Effective action in Fμν and its derivatives: NABI

Back

S = S(2) + S(4)+S(5) + · · · = 1 8πgs

• d8x

Tr

• F2

(2π)4α′2 − t8Tr

• F4

3(2π)2 +α′L(5)

• F, DF
• + · · ·
• ◮ Contributions of higher order in α′, whose rôle will be

discussed later

Effective action on the D7

(tree level)

◮ Effective action in Fμν and its derivatives: NABI

Back

S = S(2)+S(4)+S(5) + · · · = 1 8πgs

• d8x

Tr

• F2

(2π)4α′2 − t8Tr

• F4

3(2π)2 +α′L(5)

• F, DF
• + · · ·
• ◮ The quartic term has a dimensionless coupling:

S(4) = − 1 96π3gs

• d8x t8Tr
• F4

Effective action on the D7

(tree level)

◮ Effective action in Fμν and its derivatives: NABI

Back

S = S(2)+S(4)+S(5) + · · · = 1 8πgs

• d8x

Tr

• F2

(2π)4α′2 − t8Tr

• F4

3(2π)2 +α′L(5)

• F, DF
• + · · ·
• ◮ Adding the WZ term, we can write

S(4) = − 1 4!4π3gs

• d8x t8Tr
• F4

− 2πiC0 c(4) where c(4) is the fourth Chern number c(4) = 1 4!(2π)4

• Tr
• F ∧ F ∧ F ∧ F

Effective action on the D7

(tree level)

◮ Effective action in Fμν and its derivatives: NABI

Back

S = S(2)+S(4)+S(5) + · · · = 1 8πgs

• d8x

Tr

• F2

(2π)4α′2 − t8Tr

• F4

3(2π)2 +α′L(5)

• F, DF
• + · · ·
• ◮ Adding the fermionic terms, can be written using the

superfield Φ(x, θ) as

S(4) = 1 (2π)4

• d8x d8θ Tr

iπ 12 τ Φ4 + c.c.

where τ = C0 +

i gs is the axion-dilaton.

Adding D-instantons

◮ Add k D-instantons. ◮ D7/D(-1) form a 1/2 BPS system with 8

ND directions

k D(-1) ◮ D(-1) classical action

Scl = k( 2π gs − 2πiC0) ≡ −2πikτ ,

◮ Coincides with the quartic action on the D7 for gauge

fields F with c(4) = k and

• d8x Tr
• t8F4

= − 1 2

• d8x Tr
• ε8F4

= − 4! 2 (2π)4 c(4)

Adding D-instantons

◮ Add k D-instantons. ◮ D7/D(-1) form a 1/2 BPS system with 8

ND directions

k D(-1) ◮ D(-1) classical action

Scl = k( 2π gs − 2πiC0) ≡ −2πikτ ,

◮ Analogous to relation with self-dual YM config.s in

D3/D(-1)

◮ Suggests relation to some 8d instanton of the quartic

action

The size of the instanton solution

(“gauge” instantons) D(-1) D3

◮ For ordinary instantons, e.g. D3/D(-1),

there are moduli w ˙

α related to the size

ρ of the instanton profile

◮ They com from the NS sector of mixed

D3/D(-1) strings

The size of the instanton solution

(“gauge” instantons) D3 D(-1)

◮ For ordinary instantons, e.g. D3/D(-1),

there are moduli w ˙

α related to the size

ρ of the instanton profile

◮ They com from the NS sector of mixed

D3/D(-1) strings

The size of the instanton solution

(“gauge” instantons) D3 ρ

◮ For ordinary instantons, e.g. D3/D(-1),

there are moduli w ˙

α related to the size

ρ of the instanton profile

◮ They com from the NS sector of mixed

D3/D(-1) strings

The size of the instanton solution

(“gauge” instantons) D3 ρ

◮ For ordinary instantons, e.g. D3/D(-1),

there are moduli w ˙

α related to the size

ρ of the instanton profile

◮ They com from the NS sector of mixed

D3/D(-1) strings

◮ The classical instanton profile arises

from mixed disks Billo et al, 2001

Ai

μ = 2ρ2 ¯

ηi

μν

xν |x|4 + . . .

(SU(2), sing. gauge, large-|x|, 2ρ2 = tr ¯ w ˙

αw ˙ α )

p ¯ w w Aμ(p)

No size for “exotic” instantons

D(-1) D7

◮ For exotic systems, like D7/D(-1), with

“more that 4 ND directions”, mixed strings have no physical bosonic moduli

No size for “exotic” instantons

D7 D(-1)

◮ For exotic systems, like D7/D(-1), with

“more that 4 ND directions”, mixed strings have no physical bosonic moduli

◮ The configuration remains pointlike.

There is no emission diagram for the gauge field

No size for “exotic” instantons

D7 D(-1)

◮ For exotic systems, like D7/D(-1), with

“more that 4 ND directions”, mixed strings have no physical bosonic moduli

◮ The configuration remains pointlike.

There is no emission diagram for the gauge field

◮ Can one still associate the D(-1) to the zero-size limit of

some classical gauge configuration on the D7’s?

Interpretation as 8d instanton solutions

Expected features

◮ A D(-1) inside the D7’s should correspond to the

zero-size limit of some “instantonic” configuration of the SO(8) gauge field such that

◮ has 4-th Chern number c(4) = 1 ◮ the quartic action reduces to the D(-1) action, which

requires Tr(t8F4) = − 1 2 Tr(ε8F4)

◮ preserves SO(8) “Lorentz” invariance ◮ corresponds to a 1/2 BPS config. in susy case

The SO(8) instanton

◮ All our requirements met by the SO(8) instanton

[Grossmann e al, 1985]

• Aμ(x)

αβ = (γμν)αβ xν r2 + ρ2 with ρ = instanton size and r2 = xμxμ, while αβ ∈ adjoint

• f the SO(8) gauge group.

◮ is “self-dual” in the sense that F ∧ F = (F ∧ F)∗ ◮ satisfies t8F4 = −1/2ε8F4 from Clifford Algebra ◮ has c(4) = 1 and S(4) = −2πiτ

The SO(8) instanton

◮ All our requirements met by the SO(8) instanton

[Grossmann e al, 1985]

• Aμ(x)

αβ = (γμν)αβ xν r2 + ρ2 with ρ = instanton size and r2 = xμxμ, while αβ ∈ adjoint

• f the SO(8) gauge group.

◮ is “self-dual” in the sense that F ∧ F = (F ∧ F)∗ ◮ satisfies t8F4 = −1/2ε8F4 from Clifford Algebra ◮ has c(4) = 1 and S(4) = −2πiτ

◮ However, it is not a solution of Y

.M. e.o.m. in d = 8, for ρ = 0:

DμFμν(x) = 4(d − 4)ρ2 (r2 + ρ2)3 γμνxν .

Consistency conditions

◮ Eff. action on the D7 is the NABI action

Recall

◮ T

• keep the quartic action and the instanton effects the

field-theory limit must be α′ → 0 , gs fixed

Consistency conditions

◮ Eff. action on the D7 is the NABI action

Recall

◮ T

• keep the quartic action and the instanton effects the

field-theory limit must be α′ → 0 , gs fixed

◮ This limit is dangerous on the YM action SYM since

g2

YM ∝ gsα′2. On the SO(8) instanton, however, we have

(R regulates the volume):

SYM → ρ4 α′2 gs log

• ρ/R
• ,

which vanishes in the zero-size limit ρ → 0 if ρ2/α′2 → 0 (done before removing R)

Consistency conditions

◮ Eff. action on the D7 is the NABI action

Recall

◮ T

• keep the quartic action and the instanton effects the

field-theory limit must be α′ → 0 , gs fixed

◮ Consider the higher order α′ corrections to the NABI

• action. On the SO(8) instanton, by dimensional reasons,

must be ρd−8

∞

• n=1

an α′ ρ2 n ,

◮ The coefficients an should vanish for consistency!

O(F5) terms in the NABI

◮ The first coefficient a1 arises from the integral of

L(5)(F, DF), i.e.the term of order α′3 w.r.t to the YM action.

◮ We would like to check that it vanishes. Crucial point:

which is the form of L(5)(F, DF)?

O(F5) terms in the NABI

◮ The first coefficient a1 arises from the integral of

L(5)(F, DF), i.e.the term of order α′3 w.r.t to the YM action.

◮ We would like to check that it vanishes. Crucial point:

which is the form of L(5)(F, DF)?

◮ Various proposals in the literature

Refolli et al, Koerber-Sevrin, Grasso, Barreiro-Medina, ... ◮ obtained by different methods ◮ differing by terms which vanish “on-shell”, i.e. upon use of

the YM e.o.m.

O(F5) terms in the NABI

◮ The first coefficient a1 arises from the integral of

L(5)(F, DF), i.e.the term of order α′3 w.r.t to the YM action.

◮ We would like to check that it vanishes. Crucial point:

which is the form of L(5)(F, DF)?

◮ Various proposals in the literature

Refolli et al, Koerber-Sevrin, Grasso, Barreiro-Medina, ... ◮ obtained by different methods ◮ differing by terms which vanish “on-shell”, i.e. upon use of

the YM e.o.m.

◮ One proposal is singled out by admitting a susy

extension Collinucci et al, 2002

Check at O(F5) in the NABI

◮ The bosonic part of the supersimmetrizable O(α′3)

lagrangian is

L(5) = ζ(3) 2 Tr

• 4
• Fμ1μ2, Fμ3μ4
• Fμ1μ3, Fμ2μ5
• , Fμ4μ5
• +

2

• Fμ1μ2, Fμ3μ4
• Fμ1μ2, Fμ3μ5
• , Fμ4μ5
• +

2

• Fμ1μ2, Dμ5Fμ1μ4
• Dμ5Fμ2μ3, Fμ3μ4
• −

2

• Fμ1μ2, Dμ4Fμ3μ5
• Dμ4Fμ2μ5, Fμ1μ3
• +
• Fμ1μ2, Dμ5Fμ3μ4
• Dμ5Fμ1μ2, Fμ3μ4

Check at O(F5) in the NABI

◮ Plugging the instanton profile into α′

gs

• d8x L(5) we get [Using

the CADABRA program by Kasper Peeters]

α′ ζ(3) gs 2d/2+9 πd/2 Γ(9 − d/2) 9! ρ10−d × (d − 1)(d − 2)(d − 4)

• −d
• 9 −

d 2

• +
• d + 2

d 2

• namely a result proportional to

d(d − 1)(d − 2)(d − 4)(d − 8)

◮ The quintic action vanishes on the SO(8) instanton! The

check is successful

The effective action

1-loop effective action

◮ At 1-loop we get contributions from annuli and Möbius

• diagrams. At the quartic level,

F F = 0 → (Tr(F2)2) F F F F = 0 for SO(8) F F F F + F F

S1−loop

(4)

= 1 256π4

• d8x log
• Imτ ImU |η(U)|4

t8

• TrF22

= 1 (2π)4

• d8x d8θ

1 32 log

• Imτ ImU |η(U)|4

TrΦ22 + c.c. (U is the complex structure of the 2-torus T2)

Effective action from D-instantons

Instanton moduli

D7/D(-1) 8 D7-branes D(-1)/D(-1) k D-instantons

◮ Open strings with at least

• ne end on a D(–1) carry no

momentum: they are moduli rather than dynamical fields.

◮ Need to determine their

spectrum

◮ Moduli interactions via

disk diagrams encoded in Sinst

◮ D7/D7 gauge fields

interact with moduli through mixed disks

M λ a χ μ μ μ μ Φ , ... ...

Effective action from D-instantons

The idea

◮ Effective interactions between gauge fields can be

mediated by D-instanton moduli through mixed disks

• =

F F F F F F F F

◮ In the r.h.s, integrate over the moduli with a weight

exp(Sinst)

Effective action from D-instantons

Field-dependent moduli action

◮ The effective interactions for the gauge multiplet Φ can

be summarized by shifting the moduli action with Φ-dependent terms arising from mixed disks

Λ μ μ θ ϕ μ μ F μ μ θ θ

◮ In fact, we write the instanton action as

Sinst = −2πiτk + S(M(k), Φ)

Effective action from D-instantons

Field-dependent moduli action

◮ The effective interactions for the gauge multiplet Φ can

be summarized by shifting the moduli action with Φ-dependent terms arising from mixed disks

Λ μ μ θ ϕ μ μ F μ μ θ θ

◮ In fact, we write the instanton action as

Sinst = −2πiτk + S(M(k), Φ)

◮ Classical value

Effective action from D-instantons

Field-dependent moduli action

◮ The effective interactions for the gauge multiplet Φ can

be summarized by shifting the moduli action with Φ-dependent terms arising from mixed disks

Λ μ μ θ ϕ μ μ F μ μ θ θ

◮ In fact, we write the instanton action as

Sinst = −2πiτk + S(M(k), Φ)

◮ Disk interactions

Effective action from D-instantons

Moduli integral

◮ Non-perturbative contributions to the effective action of

the gauge degrees of freedom Φ arise integrating over the instanton moduli M(k) and summing over all instanton numbers k Sn.p.(Φ) =

• k

e2πiτk

• dM(k) e−S(M(k),Φ)

Effective action from D-instantons

Moduli integral

◮ Non-perturbative contributions to the effective action of

the gauge degrees of freedom Φ arise integrating over the instanton moduli M(k) and summing over all instanton numbers k Sn.p.(Φ) =

• k

e2πiτk

• dM(k) e−S(M(k),Φ)

◮ This procedure is by now well-established for instantonic

brane systems corresponding to gauge instantons

Polchinksi, 1994; Green-Gutperle, 2000, ...; T urin/Rome/Münich/UPenn/Madrid,...

Effective action from D-instantons

Moduli integral

◮ Non-perturbative contributions to the effective action of

the gauge degrees of freedom Φ arise integrating over the instanton moduli M(k) and summing over all instanton numbers k Sn.p.(Φ) =

• k

e2πiτk

• dM(k) e−S(M(k),Φ)

◮ This procedure is by now well-established for instantonic

brane systems corresponding to gauge instantons

Polchinksi, 1994; Green-Gutperle, 2000, ...; T urin/Rome/Münich/UPenn/Madrid,...

◮ We want to apply it explicitly in our “exotic” instanton

set-up

Effective action from D-instantons

Moduli integral

◮ Non-perturbative contributions to the effective action of

the gauge degrees of freedom Φ arise integrating over the instanton moduli M(k) and summing over all instanton numbers k Sn.p.(Φ) =

• k

e2πiτk

• dM(k) e−S(M(k),Φ)

◮ This procedure is by now well-established for instantonic

brane systems corresponding to gauge instantons

Polchinksi, 1994; Green-Gutperle, 2000, ...; T urin/Rome/Münich/UPenn/Madrid,...

◮ We want to apply it explicitly in our “exotic” instanton

set-up

◮ This is a very complicated matrix integral ...

The moduli spectrum

Spectrum:

Sector Name Meaning Chan-Paton Dimension −1/ − 1 NS aμ centers symm SO(k) (length) χ, ¯ χ adj SO(k) (length)−1 Dm

• Lagr. mult.

adj SO(k) (length)−2 R Mα partners symm SO(k) (length)

1 2

λ ˙

α

• Lagr. mult.

adj SO(k) (length)− 3

2

−1/7 R μ 8 × k (length) NS w (auxiliary) 8 × k (length)0

The moduli spectrum

Spectrum:

Sector Name Meaning Chan-Paton Dimension −1/ − 1 NS aμ centers symm SO(k) (length) χ, ¯ χ adj SO(k) (length)−1 Dm

• Lagr. mult.

adj SO(k) (length)−2 R Mα partners symm SO(k) (length)

1 2

λ ˙

α

• Lagr. mult.

adj SO(k) (length)− 3

2

−1/7 R μ 8 × k (length) NS w (auxiliary) 8 × k (length)0

◮ The SO(k) rep. is determined by the orientifold projection

The moduli spectrum

Spectrum:

Sector Name Meaning Chan-Paton Dimension −1/ − 1 NS aμ centers symm SO(k) (length) χ, ¯ χ adj SO(k) (length)−1 Dm

• Lagr. mult.

adj SO(k) (length)−2 R Mα partners symm SO(k) (length)

1 2

λ ˙

α

• Lagr. mult.

adj SO(k) (length)− 3

2

−1/7 R μ 8 × k (length) NS w (auxiliary) 8 × k (length)0

◮ Abelian part of aμ, Mα ∼ Goldstone modes of the

(super)translations on the D7 broken by D(-1)’s. Identified with coordinates xμ,θα

The moduli spectrum

Spectrum:

Sector Name Meaning Chan-Paton Dimension −1/ − 1 NS aμ centers symm SO(k) (length) χ, ¯ χ adj SO(k) (length)−1 Dm

• Lagr. mult.

adj SO(k) (length)−2 R Mα partners symm SO(k) (length)

1 2

λ ˙

α

• Lagr. mult.

adj SO(k) (length)− 3

2

−1/7 R μ 8 × k (length) NS w (auxiliary) 8 × k (length)0

◮ For “mixed” strings, no bosonic moduli from the NS

sector: characteristic of “exotic” instantons

The moduli action

◮ The action reads:

S(M(k), Φ) = tr

• i λ ˙

αγ ˙ αβ μ [aμ, Mβ] +

1 2g2 λ ˙

α[χ, λ ˙ α] + Mα[¯

χ, Mα] + 1 2g2 DmDm − 1 2 Dm(τm)μν aμ, aν +

• aμ, ¯

χ

• [aμ, χ] +

1 2g2 [¯ χ, χ]2 + tr

• μT μ χ
• + tr
• μT Φ(x, θ) μ
• +tr
• wT w
• ◮ The “supercoordinate” moduli x, θ only appear through

Φ(x, θ). The remaining “centred” moduli are denoted as

• M(k)

All instanton numbers ...

... lead to quartic terms

◮ Effective action (using q = e2πiτ):

Sn.p.(Φ) =

• d8x d8θ
• k

qk

• d

M(k) e−S(

M(k),Φ(x,θ))

◮ In our “conformal” set-up, with with SO(8) gauge group

• n the D7, counting the dimensions of the moduli we get
• d

M(k)

• = (length)−4

◮ Thus

d M(k) e−S(

M(k),Φ(x,θ)) = quartic invariant in Φ(x, θ) ◮ Integration over d8θ leads to terms of the form “t8 F4 ”

All instanton numbers ...

... lead to quartic terms

◮ Effective action (using q = e2πiτ):

Sn.p.(Φ) =

• d8x d8θ
• k

qk

• d

M(k) e−S(

M(k),Φ(x,θ))

◮ In our “conformal” set-up, with with SO(8) gauge group

• n the D7, counting the dimensions of the moduli we get
• d

M(k)

• = (length)−4

◮ Thus

d M(k) e−S(

M(k),Φ(x,θ)) = quartic invariant in Φ(x, θ) ◮ Integration over d8θ leads to terms of the form “t8 F4 ” ◮ The “non-conformal” case of N = 8 D7’s has been

considered in Fucito et al, 2009

One-instanton case

◮ For k = 1 things are particularly simple

◮ The spectrum of moduli is reduced to {x, θ, μ} ◮ The moduli action is simply Sinst = −2πiτ + μT Φ(x, θ) μ

◮ The instanton-induced interactions are thus

• d8x d8θ q
• dμ e−μT Φ(x,θ) μ ∼
• d8x d8θ qPf
• Φ(x, θ)
• ◮ A new structure, associated to the SO(8) invariant

“ t8Pf(F) ”, appears in the effective action at the

• ne-instanton level after the d8θ integration

Multi-instantons

◮ For k > 1 things are more complicated, but we can

exploit the SUSY properties of the moduli action, which lead to:

◮ an equivariant cohomological BRST structure ◮ a localization of the moduli integrals (after suitable closed

string deformations)

◮ Similar techniques have been successfully used to

◮ compute the YM integrals in d = 10, 6, 4 and the

D-instanton partition function

Moore+Nekrasov+Shatashvili, 1998 ◮ compute multi-instanton effects in N = 2 SYM in d = 4 and

compare with the Seiberg-Witten solution

Nekrasov, 2002; + ... ◮ derive the multi-instanton calculus using D3/D(–1) brane

systems

Fucito et al, 2004; Billò et al, 2006; ...

Deformations from RR background

◮ Suitable deformations that help to fully localize the

integral arise from RR field-strengths 3-form with one index on T2 Fμν ≡ Fμνz , ¯ Fμν ≡ Fμνz

◮ The Fμν is taken in an SO(7)⊂ SO(8) (Lorentz) with

spinorial embedding

◮ Disk diagrams with RR insertions

modify the moduli action

S( M(k), φ) → S( M(k), φ, F)

(here we introduced the v.e.v. φ = 〈Φ〉)

F λ λ

BRST structure

Equivariance

◮ Single out one of the supercharges Q ˙

α, say Q = Q8. After

relabeling some of the moduli:

Mα → Mμ ≡ (Mm, −M8) , λ ˙

α → (λm, η) ≡ (λm, λ8)

• ne has

Qaμ = Mμ , Qλm = −Dm , Q¯ χ = −i

• 2η , Qχ = 0 , Qμ = w

◮ Moreover, on any modulus,

Q2 • = TSO(k)(χ)• + TSO(8)(φ)• + TSO(7)(F)•

where

◮ TSO(k)(χ) = inf.mal SO(k) rotation parametrized by χ ◮ TSO(8)(φ) = inf.mal SO(8) rotation parametrized by φ ◮ TSO(7)(F) = inf.mal SO(7) rotation parametrized by F

Symmetries of the moduli

◮ The action of the BRS charge Q is thus determined by

the symmetry properties of the moduli

SO(k) SO(7) SO(8) aμ symm 8s 1 Mμ symm 8s 1 Dm adj 7 1 λm adj 7 1 ¯ χ adj 1 1 η adj 1 1 χ adj 1 1 μ k 1 8v

BRST structure

Exactness

◮ The (deformed) action is BRST-exact:

S( M(k), φ, F) = QΞ

◮ ¯

F only appears in the “gauge fermion” Ξ: the final result does not depend on it

◮ The (deformed) BRST structure allows to suitably rescale

the integration variables and show that the semiclassical approximation is exact

Moore+Nekrasov+Shatashvili, 1998; ...; Nekrasov, 2002; Flume+Poghossian, 2002; Bruzzo et al, 2003; ...

Scaling to localization

◮ Many integrations reduce to quadratic forms:

Zk(φ, F) ≡

• dM(k) e−S(

M(k),φ,F)

= ... = ... =

• {dadMdDdλ dμ dχ} e−tr{ g

2 D2− g 2 λ

Q2λ+ t

4 a

Q2a+ t

4 M2+tμ

Q2μ}

∼

• {dχ}

Pfλ

• Q2

Pfμ

• Q2

deta

• Q21/2

◮ The χ integrals can be done as contour integrals and the

final result for Zk(φ, F) comes from a sum over residues

Moore+Nekrasov+Shatashvili, 1998

The recipe

◮ From the explicit expression of Zk(φ, F), we can obtain

the non-perturbative effective action. However:

◮ At instanton number k, there are disconnected

contributions from smaller instantons ki (with

• i ki = k).

T

• isolate the connected components we have to take

the log:

Z =

• k

Zk(φ, F)qk → logZ

◮ In obtaining Zk(φ, F) we integrated also over x and θ

producing a factor of E−1 ∼ det(F)−1/2. T

• remove this

contribution we have to multiply by E

logZ → E logZ

before turning off the RR deformation.

The prepotential

◮ All in all, we obtain the non-perturbative part of the

D7-brane effective action: S(n.p.) = 1 (2π)4

• d8x d8θ F(n.p.)
• Φ(x, θ)
• with the “prepotential” F(n.p.)(Φ) given by

F(n.p.)(Φ) = E logZ

• φ→Φ,F→0

and with

Z =

• k

Zk(φ, F)qk , E ∼ det(F)1/2

Explicit results

◮ Expanding in instanton numbers, F(n.p.) =

• k qkFk, we

have

F1 = EZ1 , F2 = EZ2 − F2

1

2E , F3 = EZ3 − F2F1 E − F3

1

6E2 F4 = EZ4 − F3F1 E − F2

2

2E − F2F2

1

2E2 − F4

1

24E3 , F5 = EZ5 − F4F1 E − F3F2 E − F3F2

1

2E2 − F2

2F1

2E2 − F2F3

1

6E3 − F5

1

120E4 , . . . . . .

Explicit results

◮ Expanding in instanton numbers, F(n.p.) =

• k qkFk, we

have

F1 = 8Pf(Φ) , F2= EZ2 − F2

1

2E , F3= EZ3 − F2F1 E − F3

1

6E2 , F4= EZ4 − F3F1 E − F2

2

2E − F2F2

1

2E2 − F4

1

24E3 , F5= EZ5 − F4F1 E − F3F2 E − F3F2

1

2E2 − F2

2F1

2E2 − F2F3

1

6E3 − F5

1

120E4 , . . . . . .

Explicit results

◮ Expanding in instanton numbers, F(n.p.) =

• k qkFk, we

have

F1 = 8Pf(Φ) , F2 = 1 2 TrΦ4 − 1 4

• TrΦ22 ,

F3= EZ3 − F2F1 E − F3

1

6E2 , F4= EZ4 − F3F1 E − F2

2

2E − F2F2

1

2E2 − F4

1

24E3 , F5= EZ5 − F4F1 E − F3F2 E − F3F2

1

2E2 − F2

2F1

2E2 − F2F3

1

6E3 − F5

1

120E4 , . . . . . .

Explicit results

◮ Expanding in instanton numbers, F(n.p.) =

• k qkFk, we

have

F1 = 8Pf(Φ) , F2 = 1 2 TrΦ4 − 1 4

• TrΦ22 ,

F3 = 32 3 Pf(Φ) , F4= EZ4 − F3F1 E − F2

2

2E − F2F2

1

2E2 − F4

1

24E3 , F5= EZ5 − F4F1 E − F3F2 E − F3F2

1

2E2 − F2

2F1

2E2 − F2F3

1

6E3 − F5

1

120E4 , . . . . . .

Explicit results

◮ Expanding in instanton numbers, F(n.p.) =

• k qkFk, we

have

F1 = 8Pf(Φ) , F2 = 1 2 TrΦ4 − 1 4

• TrΦ22 ,

F3 = 32 3 Pf(Φ) , F4 = 1 4 TrΦ4 − 1 4

• TrΦ22 ,

F5 = 48 5 Pf(Φ) , . . . . . .

Explicit results

◮ The D-instanton induced effective “prepotential” is

F(n.p.)(Φ) =8Pf(Φ)

• q +

4 3 q3 + 6 5 q5 + . . .

• + TrΦ4

1 2 q2 + 1 4 q4 + . . .

• +
• TrΦ22

1 4 q2 + 1 4 q4 + . . .

• ◮ It is natural to generalize these results and write

F(n.p.)(Φ) =8Pf(Φ)

• k=1

d2k−1 q2k−1 + 1 2 TrΦ4

k=1

• dk q2k − dk q4k

+ 1 8

• TrΦ22

k=1

• dk q4k − 2dk q2k

with dk =

• ℓ|k

1 ℓ sum over the inverse divisors of k

Complete result

◮ T

aking into account the contributions at tree-level for TrF4 and at 1-loop for

• TrF22, the full expression for the

quartic terms in the effective action of the D7-branes reads

2t8 Pf(F) log

• η(τ + 1/2)

η(τ)

• 4

+ t8 TrF4 4 log

• η(4τ)

η(2τ)

• 4

+ t8 (TrF2)2 16 log

• Im τ ImU

|η(2τ)|8 |η(U)|4 |η(4τ)|4

• with q = e2πiτ

Heterotic / T ype I′ duality

◮ In the SO(8)4 Heterotic String on

T2 the BPS-saturated quartic terms in F arise at 1-loop

t8 TrF4 4 log

• η(4T)

η(2T)

• 4

+ t8 (TrF2)2 16 log

• ImT ImU

|η(2T)|8 |η(U)|4 |η(4T)|4

• Lerche+Stieberger, 1998; Gutperle, 1999; Kiritsis et al, 2000; ...

+2t8 Pf(F) log

• η(T + 1/2)

η(T)

• 4

Gava et al, 1999

◮ Agrees with our T

ype I’ result under the duality map T : Kähler structure of the 2-torus T2 ←→ τ : axion-dilaton world-sheet instantons ←→ D-instantons

Conclusions and perspectives

Remarks

◮ We have explicitly computed the effective couplings

induced by stringy instantons in a simple 8d example, the D7/D(–1) system in T ype I′, extending the philosophy used for “ordinary” instantons

◮ If we do not switch off the RR background F in the final

expressions we get also non-perturbative gravitational corrections to TrR4 and TrR2TrF2

Remarks

◮ We have explicitly computed the effective couplings

induced by stringy instantons in a simple 8d example, the D7/D(–1) system in T ype I′, extending the philosophy used for “ordinary” instantons

◮ If we do not switch off the RR background F in the final

expressions we get also non-perturbative gravitational corrections to TrR4 and TrR2TrF2

◮ The result checks out perfectly against the dual Heterotic

SO(8) theory:

◮ Assuming the duality, confirms our procedure to deal with

the stringy instantons

◮ Assuming the correctness of our computation, yields very

non-trivial check of this fundamental string duality

4d exotic instanton calculus

◮ We’re now investigating simple models where

◮ the gauge theory lives in four dimensions ◮ there are “exotic” instantons, with no “size” moduli from

mixed strings having more than 4 ND directions, ...

◮ all instanton numbers can contribute to the effective

action at order F2 (“conformal” situation)

◮ there is the chance of checking the result against a dual

heterotic theory

A specific model

◮ In particular, we are considering T

ype I’ theory on R1,3 × T4/Z2 × T2 with 8 D7-branes and 8 D3-branes (T-dual on T2 of PS-GP model)

◮ D(-1)’s represent exotic instantons w.r.t. to the gauge

theory on the D7’s

◮ Preliminary analysis indicates that the calculus of the

induced effective action is feasible with methods analogues to those presented here